The Reduced Basis Method:

Transcription

1 The Reduced Basis Method: Application to Problems in Fluid and Solid Mechanics ++ Aachen Institute for Advanced Study in Computational Engineering Science K. Veroy-Grepl 31 January 2014

2 AICES & MOR I The AICES Graduate School was established in 2006 as part of the Excellence Initiative I Interdisciplinary research; collaborative e ort among 28 institutes from 8 departments I Substantial activity in MOR AC.CES Conference (2015) EU Regional School Prof. Dr. Peter Benner MPI for Dynamics of Complex Technical Systems Magdeburg Short course: 6-May, 14:00 17:30 COST Action EU-MORNET Spearheaded by Prof. Wil Schilders (TU Eindhoven) 1/32

3 Collaborators Eduard Bader Lorenzo Zanon Zhenying Zhang Mark Kärcher Also A.-L. Gerner M. Grepl Y. Maday A.T. Patera C. Prud homme D. Rovas 2/32

4 The Reduced Basis Method The RB Method provides rapid and reliable approximation of solutions to parametrized partial di erential equations (µpdes) for design and optimization, control, parameter estimation,... parameter i.e.,space exploration. ^ 3/32

5 The Reduced Basis Method SNAPSHOTS u(µ i ) EXACT SOLUTION u(µ) u N (µ) APPROXIMATION ERROR BOUND N(µ) HIGH-DIMENSIONAL FE SPACE * OFFLINE-ONLINE COMPUTATIONAL DECOMPOSITION * GREEDY ALGORITHM (see, e.g., [PRUD HOMME, ROVAS, V., MACHIELS, MADAY, PATERA, & TURINICI, 2002], [V., PRUD HOMME, ROVAS, & PATERA, 2003]) 4/32

6 RB Method for Problems in Solid Mechanics Goal Issues Develop e cient and reliable reduced order models for parametrized nonlinear problems in solid mechanics. I Main di culty is nonlinearity which appears in many forms Buckling Hyperelastic materials relevant, for example, in modeling of biological tissues Contact problems constrained problems, contact area unknown a priori Plasticity nonlinear material laws, yield area unknown a priori, irreversibility, rate dependence... 5/32

7 RB Method for Problems in Solid Mechanics Buckling I Find critical load parameter at which solution becomes non-unique. FE SPACE X A v = B(u)v 6/34

8 RB Method for Problems in Solid Mechanics Initial Results 10 2 Linear problem: Error for different values of N Eigenvalue problem: Error for different parameters Parameter, µ N [with ZANON (in progress)] 7/34

9 RB Method for Problems in Solid Mechanics Hyperelastic materials I POD-based model order reduction for finite deformation problems [RADERMACHER &REESE, 2013] I Empirical Interpolation Method or EIM (and variants thereof) for approximation of non-a ne / nonlinear terms [BARRAULT, MADAY, NGUYEN &PATERA, 2004] [GREPL, MADAY, NGUYEN &PATERA, 2007] I Apply EIM along with the RB method for hyperelasticity [with ZANON (in progress)] 8/32

10 RB Method for Problems in Solid Mechanics Contact (and Plasticity) I RB method for a 1-D variational inequality, but error estimates require high-dimensional computations [HAASDONK, SALOMON &WOHLMUTH, 2012] I Penalty and barrier techniques to transform nonlinearity due to the constraint into tractable nonlinear term I EIM for approximation of the nonlinear terms [with BADER &ZHANG (in progress)] [GREPL, MADAY, NGUYEN &PATERA, 2007] 9/32

11 RB Method for Problems in Solid Mechanics Initial Results Obstacle problem Relative Error and Error Estimates M = 10 true M = 10 estim. M = 15 true M = 15 estim. M = 20 true M = 20 estim x N [with BADER &ZHANG (in progress)] 10/34

12 RB Method for Fluid Flow Goal Develop e cient and reliable reduced order models for incompressible fluid flow problems in parametrized geometries. Issues I Nonlinearity in the Navier-Stokes equations I Even in the simplest case (Stokes), the saddle point problem structure is more di cult: A u = f vs. apple A B B T 0 apple u p = apple f g 11/32

13 Incompressible Navier-Stokes Equations Momentum Equations a 0 (u, v) + a {z } 1 (u, u, v) {z } di usion convection + b(v, p) {z } pressure term = f(v), 8 v 2X {z } forcing Continuity Equation b(u, q) = 0, 8 q 2Y I Nonlinear Problem I Saddle-Point Problem 12/34

14 Parametrized Navier-Stokes Motivation Model Problem Incompressible Navier-Stokes Equations Parametrized Navier-Stokes Motivation Example: Natural Convection in a Cavity Natural Convection in a Cavity Model Problem I (u1(µ), u2(µ), T (µ), p(µ)) x 2, u2 101 Gr = 1.0 (A = 4, 1) gravity (0, 0) x 1, u1 Gr = (say, µ (Gr, Pr)) Parameters: µ = (Gr, Pr = 0) Field variables: (u(µ), p(µ), T (µ)) [Gamm], [SRC], [GBY]. Gr = In the absence of geometric parameters the bilinear form b(, ) is Solutions at Pr = 0, Note the one-roll to three-roll transition is nonsingular [SRC]. parameter-independent. Gr = {101, 4 104, 105 } [V. & PATERA, 2005] 13/32

15 Incompressible Navier-Stokes Equations Observation I Given set of parameter values µ i 2D,i=1,...,n, and: Y n := span{ p(µ i ), i =1,...,n } If X n is the space of velocity snapshots X n := span { u(µ i ), i =1,...,n } PRESSURE VELOCITY then any v 2X n satisfies b(v, q) =0for all q 2Y n Y Need only consider a 0 (u, v; µ) +a 1 (u, u, v; µ) = f(v; µ), 8 v 2X 0 where X 0 = {v 2X r v =0} X n. 14/32

16 Incompressible Navier-Stokes Equations FE SPACE X I Smooth solution manifold I Possibility of bifurcation or multiple solution branches I Focus on an isolated, non-singular solution branch I Compute Galerkin projection onto space spanned by snapshots Issues: Error Estimation and O ine-online Decomposition I Rigorous treatment feasible for quadratic (and cubic) nonlinearities I For more general nonlinearities: Empirical Interpolation Method 15/34

17 Incompressible Navier-Stokes Equations A Posteriori Error Estimation N(µ)/ (µ), N(µ), N(µ) u(µ) u N (µ) N(µ) N(µ)/ N(µ)? u(µ) other branches IF N (µ) N (µ)/ N(µ) 2 < 1 [V. & PATERA, 2005] 16/32

18 Incompressible Navier-Stokes Equations Natural Convection Example n MAX RELATIVE ERROR PROXIMITY ku u nk X /kuk INDICATOR X n MAX RELATIVE ERROR BOUND n/kuk X EFFEC- TIVITY n E E E E E E E E E E E E E E E I For µ =Gr2 [1, 10 5 ],andpr = 0 I Recall that n must be < 1 to obtain bounds. [V. & PATERA, 2005] 17/32

19 Saddle Point Problem Stokes Equations a(u, v; µ) +b(v, p; µ) = f(v; µ), 8 v 2X b(u, q; µ) = 0, 8 q 2Y Reduced-Basis Equations a(u n,v; µ) +b(v, p n ; µ) = f(v; µ), 8 v 2X n b(u n,q; µ) = 0, 8 q 2Y n Issues I Approximation: Construction of good, stable spaces X n and Y n I Error Estimation: Development of e ciently computable bounds 18/32

20 RB Method for Fluid Flow Example Stokes flow in a channel with rectangular obstacle x2 0(µ) (A =4, 1) in µ1 out (0, 0) MESH DOFS µ2 0(µ) 16,600 elements 72,000 (total) (µ) x1 [GERNER &V.,2012] 19/32

21 Saddle Point: Approximation Stability The RB spaces X n, Y n constitute a stable pair if for all µ 2D b(v, q; µ) n(µ) inf sup > 0 q2y n v2x n kvk X kqk Y Pressure Space For µ i 2D,i=1,...,n, and Reduced Basis Error vs n velocity pressure Y n := span{p(µ i ), i =1to n} 10 1 Velocity Space I Option 0: The Naive Choice 10 2 X 0 n := span{u(µ i), i =1to n} N 20/32

22 Saddle Point: Approximation Velocity Space I Option 1 ) provably stable Xn 1 := span{u(µ i), T µ p(µ i ) } {z } SUPREMIZERS I Option 2 ) justifiably stable X 2 n := span{u(µ i),t µi p(µ i )} I Option 3 ) empirically stable X 3 n := span{u(µ i),u(µ 0 i )} [ROVAS, 2003], [ROZZA &V.,2007] [GERNER &V.,2012] Velocity error vs n 10 0 total Option 1 Option Option Pressure error vs n 10 0 total Option 1 Option Option N Z 21/32

23 Saddle Point: Error Estimation 1. Treat entire system as a general noncoercive problem A(U(µ),V; µ) =F(V ; µ), 8 V 2Z Letting R(V ; µ) denote the residual, we have (Ban-Neč-Bab) ku(µ) U n(µ)k Z apple kr( ; µ)k Z 0 A LB (µ) =: N (µ) [V., PRUD HOMME, ROVAS, & PATERA, 2003] [ROZZA, etal.] 2. Treat the system as a saddle point problem (Brezzi) ku u n k X apple kr1 N k X u0 UB + 1+ a u =: a LB a N LB kp p n k X apple kr1 N k X u0 b LB assuming a is coercive. kr 2 N k X p0 b LB UB + a b ke u N k X u =: p N LB (µ) [GERNER &V.,2012] 22/32

24 Saddle Point: Error Estimation Velocity Error Bound 10 2 e u / u u / u Ba / u Pressure Error Bound e p / p p / p Ba / p N Z 23/32

25 RB Method for Optimal Control Goal E Issues cient and reliable solution of parametrized PDE-constrained optimal control problems. I Find control input such that the system achieves a desired state: STATE CONTROL u = arg min u J( y, u ) {z } COST where Ay =fu {z } PDE where all quantities are parameter-dependent. I Use of reduced order model as surrogate leads to errors in the optimal control input and the cost function 24/32

26 RB Method for Optimal Control Status I Perturbation approach sharp error bounds for POD, but requires high-dimensional computations [TRÖLTZSCH &VOLKWEIN, 2009] e ciently computable error bounds for RB but e ectivities tend to be large for small I Banach-Nečas-Babuška approach combined control-state bounds, required constants di [KAERCHER &GREPL, 2013] cult to compute [NEGRI, MANZONI, ROZZA, &QUARTERONI, 2013] 25/32

27 RB Method for Optimal Control Recent Contribution I Derived alternative error bound through manipulation of the error-residual equation for the optimality system. I Rigorous error bounds depend only on (relatively) easily computable constants I Comparison with other approaches reveal superiority of the bound insharpness,e ciency,aswellas insensitivity to the regularization parameter [KAERCHER, GREPL, &V.2014(inprep)] 26/32

28 RB Method for Optimal Control Control of Temperature Distribution x (µ 1,µ 2 ) 3,apple ,apple 2 1,apple x 1 I Parameter-dependent steady heat conduction I Cost function J(y, u) = 1 2 ky y dk 2 + ku u d k /32

29 RB Method for Optimal Control Control of Temperature Distribution Control State 28/32

30 RB Method for Optimal Control Some Results Max. rel. Control Error and Bound KGV Per BNB Error Average Effectivities of Control Error Bound KGV Per BNB TV M = N/ regularization parameter λ [KAERCHER, GREPL &V.,2014(inprep)] 29/32

31 RB Method for Parameter Estimation Goal Reliable parameter estimation for systems governed by PDEs. s(µ) EXACT OUTPUT EXACT MEASUREMENT EXACT SOLUTION µ UNKOWN PARAMETER 30/34

32 RB Method for Parameter Estimation Goal Reliable parameter estimation for systems governed by PDEs. s(µ) M MEASUREMENT A SET OF ALL PARAMETERS WHICH AGREE WITH THE MEASUREMENT µ 30/34

33 RB Method for Parameter Estimation Goal Reliable parameter estimation for systems governed by PDEs. s(µ) REDUCED BASIS OUTPUT BOUNDS s + N(µ) s N (µ) M MEASUREMENT [s + N(µ),s N (µ)] M NON-EMPTY INTERSECTION A N ADMISSIBLE REGION µ 30/34

34 RB Method for Parameter Estimation Goal Reliable parameter estimation for systems governed by PDEs. s(µ) MULTIPLE REGIONS M MEASUREMENT A N ADMISSIBLE REGIONS µ 30/34

35 RB Method for Parameter Estimation µ 2 EXACT SOLUTION ADMISSIBLE REGION A µ 2 [GREPL, et al. (2007)] INITIAL CENTER SEARCH DIRECTIONS ADMISSIBLE REGION A N BOUNDARY POINTS µ 1 µ 1 Status I Assume that admissible region is simply-connected and convex (or star-shaped) I Use Levenberg-Marquardt algorithm to find a starting feasible point I At given directions, use bisection to find points on the boundary [GREPL, NGUYEN, V.,PATERA, &LIU, 2007] 31/34

36 RB Method for Parameter Estimation Recent Contributions I Key observations: We actually seek a curve in parameter space The level set method [OSHER &SETHIAN, 1988] is particularly suited for tracking curves and surfaces. I Level set + RB method to determine the admissible region [7] [GREPL &V.,2011] 32/32

37 RB Method for Parameter Estimation Some Results Nondestructive evaluation of (FRP) reinforced concrete MEAS 1 MEAS Measurement error 5%, N= FRP LAMINATE w apple DELAMINATION µ CONCRETE SLAB 1.05 µ * = (4,1.2) µ IC µ [GREPL &V.,2011] 33/34

38 RB Method for Parameter Estimation Some Results Nondestructive evaluation of (FRP) reinforced concrete MEAS 1 MEAS Measurement error 1%, N= FRP LAMINATE w apple DELAMINATION µ CONCRETE SLAB 1.05 µ * = (4,1.2) µ IC µ [GREPL &V.,2011] 33/34

39 RB Method for Parameter Estimation Some Results Nondestructive evaluation of (FRP) reinforced concrete MEAS 1 MEAS Measurement error 1%, N= FRP LAMINATE w apple DELAMINATION µ CONCRETE SLAB 1.05 µ * = (4,1.2) µ IC µ [GREPL &V.,2011] 33/34

40 Summary and Perspectives I For systems governed by parametrized PDEs, the RB Method o ers e cient, reliable approximations for parameter estimation, optimization, control,... I Research will continue to focus on basic methodology: e.g., nonlinear solid mechanics parametrized geometries;... but will increasingly be driven by applications: e.g., optimal design of building components, control of building climate, control of power networks? S. Reese (RWTH), et al. with D. Müller (RWTH) & C. Prud homme (Uni Strasbourg)? with A. Monti (RWTH) and M. Grepl (IGPM) 34/34

41 [plain]scientific References C. Prud homme, D.V. Rovas, K. Veroy, L. Machiels, Y. Maday, A.T. Patera, and G. Turinici, Reliable real-time solution of parametrized partial di erential equations: Reduced-basis output bound methods, J. Fluid. Eng., vol. 124, no. 1, pp , K. Veroy, Reduced-basis methods applied to problems in elasticity: Analysis and applications, Ph.D. dissertation, Massachusetts Institute of Technology, M. Barrault, Y. Maday, N.C. Nguyen, and A.T. Patera, An empirical interpolation method: application to e cient reduced-basis discretization of partial di erential equations, C. R. Math., vol. 339, no. 9, pp , B. Haasdonk, J. Salomon, and B. Wohlmuth, A reduced basis method for parametrized variational inequalities, SIAM J. Numer. Anal., vol. 50, no. 5, pp , A. Radermacher, and S. Reese, Proper orthogonal decomposition-based model reduction for nonlinear biomechanical analysis, Internat. J. Mat. Eng. Innovation, Special Issue on Computational Mechanics and Methods in Applied Materials Engineering, vol. 4, no. 4, , K. Veroy and A.T. Patera, Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds, Internat. J. Numer. Methods Fluids, vol. 47, pp , K. Veroy, C. Prud homme, D.V. Rovas, and A.T. Patera, A posteriori error bounds for reduced- basis approximation of parametrized noncoercive and nonlinear elliptic partial di erential equations, in Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, 2003, AIAA Paper G. Rozza, D.B.P. Huynh, A. Manzoni, Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants, Numerische Mathematik, vol. 125, no. 1, pp , 2013.

42 Scientific References (cont d.) G. Rozza and K. Veroy, On the stability of the reduced basis method for Stokes equations in parametrized domains, Comput. Methods Appl. Mech. Engrg., vol. 196, no. 7, pp , A.-L. Gerner and K. Veroy, Reduced basis a posteriori error bounds for the stokes equations in parametrized domains: A penalty approach, 2010, accepted in Math. Models Methods Appl. Sci. A.-L. Gerner and K. Veroy, Certified reduced basis methods for parametrized saddle point problems, SIAM J. Sci. Comput., vol. 34, no. 5, pp. A2812 A2836, M.A. Grepl, N. C. Nguyen, K. Veroy, A. T. Patera, and G. R. Liu, Certified Rapid Solution of Partial Di erential Equations for Real-Time Parameter Estimation and Optimization, L.T. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes, and B. van Bloemen Waanders, Eds. Philadephia, PA: Society for Industrial and Applied Mathematics, M.A. Grepl and K. Veroy, A level set reduced basis approach to parameter estimation, C. R. Math., vol. 349, no. 2324, pp , F. Tröltzsch and S. Volkwein, POD a posteriori error estimates for linear-quadratic optimal control problems, Comput. Optim. Appl., vol. 44, pp , F. Negri, G. Rozza, A. Manzoni, and A. Quarteroni, Reduced basis method for parametrized elliptic optimal control problems, SIAM J. Sci. Comput., 2013, to appear. M. Kaercher and M.A. Grepl. A Certified Reduced Basis Method for Parametrized Elliptic Optimal Control Problems. ESAIM: Control, Optimisation and Calculus of Variations, 2013 (accepted).

43 Thank you for your attention. Questions?